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He was the only son of James Smith, a wealthy merchant and antiquary of the Jordanhill estate, Glasgow, and his wife Mary, daughter of Alexander Wilson, professor of astronomy in Glasgow University. Archibald studied at Glasgow University in 1828, and then at Trinity College, Cambridge, where he was Senior Wrangler, said to be the first Scot to achieve this position, and first Smith's prizeman in 1836, elected a fellow of Trinity College. He was one of the founders of the Cambridge Mathematical Journal. He entered Lincoln's Inn, and was called to the bar in 1841, practising as an equity draughtsman and property lawyer.
His scientific work was mainly in the field of applications of magnetism and the Earth's magnetic field. He obtained practical formulae for the correction of magnetic compass observations made on board ship, which General Sir Edward Sabine published in the Transactions of the Royal Society: Smith later made convenient tables. In 1859 he edited William Scoresby's Journal of a Voyage to Australia for Magnetical Research and gave an exact formula for the effect of the iron of a ship on the compass. In 1862, in conjunction with the hydrographer Sir Frederick John Owen Evans FRS (1815-1885), then superintendent of the compass department of the navy, he published an Admiralty Manual for ascertaining and applying the Deviations of the Compass caused by the Iron in a Ship.
He was elected a Fellow of the Royal Society of Edinburgh in 1837. Elected a Fellow of the Royal Society in June 1856, he was awarded its Royal Medal in 1865 "for his papers in the Philosophical Transactions and elsewhere, on the magnetism of ships". In 1866 Emperor Alexander II of Russia presented him with a gold compass, set in diamonds, and emblazoned with the Imperial Arms.
In 1853, Smith married Susan Emma, daughter of Sir James Parker of Rothley Temple, Leicestershire. They had six sons and two daughters, the eldest James Parker Smith, becoming M.P. for Partick, Lanarkshire. A middle son, Henry Babington Smith, became a senior civil servant, and his fourth son, Arthur Smith (curator), became a museum curator.
- George Stewart, 'Archibald Smith', in Curiosities of Glasgow Citizenship, 1881, p. 238
- "Smith, Archibald (SMT832A)". A Cambridge Alumni Database. University of Cambridge.
- J. C. Mawell, A Treatise on Electricity and Magnetism, Volume 2, section 441.
- "Smith, Archibald". Dictionary of National Biography. London: Smith, Elder & Co. 1885–1900.
- Obituary notice by "W.T." (Lord Kelvin), Proceedings of the Royal Society 22 (1873-1874) pp. 1-xxiv  (the first known occurrence of the phrase harmonic analysis is on p.vi )